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Gauge Symmetry Restoration by Higgs Condensation in Flux Compactifications on Coset Spaces

Extra-dimensional components of gauge fields in higher-dimensional gauge theories will play a role of the Higgs field and become tachyonic after Kaluza-Klein compactifications on internal spaces with (topologically nontrivial) gauge field backgrounds. Its condensation is then expected to break gauge symmetries spontaneously. But, contrary to the expectation, some models exhibit restoration of gauge symmetries. In this paper, by considering all the massive Kaluza-Klein excitations of gauge fields, we explicitly show that some of them indeed become massless at the minimum of the Higgs potential and restore (a part of) the gauge symmetries which are broken by gauge field backgrounds. We particularly consider compactifications on $S^2$ with monopole-like fluxes and also on $\mathbb{CP}^2$ with instanton and monopole-like fluxes. In some cases, the gauge symmetry is fully restored, as argued in previous literature. In other cases, there is a stable vacuum with a partial restoration of the gauge symmetry after Higgs condensation. Topological structure of the gauge field configurations prevent the gauge symmetries to be restored.